Four random chords forming a quadrilateral

Mark $8$ points on a circle and draw every chord joining two of them. Now choose $4$ of these chords uniformly at random. What is the probability that the chosen chords are exactly the four sides of a convex quadrilateral?

$2/585$. There are $\binom{8}{2}=28$ chords, hence $\binom{28}{4}=20475$ equally likely choices. Each set of $4$ points determines exactly one convex quadrilateral, giving $\binom{8}{4}=70$ good choices, and $70/20475=2/585$.